Subgroups — #72266340

Order 4096.

At least 19838 conjugacy classes of subgroups; only the 357 normal subgroups are shown.

The subgroup lattice has 274 nodes (normal subgroups only) — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#72266334	C₂	1	1	yes	maximal, characteristic, complemented
2	#72266330	C₂	1	1	yes	maximal, characteristic, complemented
3	#72266320	C₂	1	1	yes	maximal, characteristic, complemented
4	#72266316	C₂	1	1	yes	maximal, characteristic, complemented
5	#72266301	C₂	1	1	yes	maximal, characteristic, complemented
6	#72266288	C₂²	1	1	yes	characteristic, complemented
7	#72266245	C₂	1	1	yes	maximal, characteristic, complemented
8	#72266233	C₂	1	1	yes	maximal, characteristic, complemented
9	#72266215	C₂	1	1	yes	maximal, characteristic, complemented
10	#72266201	C₂²	1	1	yes	characteristic, complemented
11	#72266194	C₂²	1	1	yes	characteristic, complemented
12	#72266177	C₂	1	1	yes	maximal, characteristic, complemented
13	#72266142	C₂	1	1	yes	maximal, characteristic, complemented
14	#72266064	C₂	1	1	yes	maximal, characteristic, complemented
15	#72265775	C₂	1	1	yes	maximal, characteristic, complemented
16	#72265766	C₂²	1	1	yes	characteristic, complemented
17	#72259019	C₂²	1	1	yes	characteristic, complemented
18	#72241156	C₂²	1	1	yes	characteristic
19	#72240423	C₂	1	1	yes	maximal, characteristic, complemented
20	C₂×#133278	C₂²	1	1	yes	characteristic
21	#72238594	C₂²	1	1	yes	characteristic
22	#72238594	C₂²	1	1	yes	characteristic
23	#72238594	C₂²	1	1	yes	characteristic, complemented
24	C₂×#6628896	C₂²	1	1	yes	characteristic
25	#72237035	C₂²	1	1	yes	characteristic, complemented
26	#72236921	C₂	1	1	yes	maximal, characteristic, complemented
27	#72236901	C₂²	1	1	yes	characteristic, complemented
28	#72236837	C₂²	1	1	yes	characteristic, complemented
29	#72236816	C₂²	1	1	yes	characteristic, complemented
30	#72236754	C₂²	1	1	yes	characteristic
31	#72236754	C₂²	1	1	yes	characteristic, complemented
32	#72235872	C₂	1	1	yes	maximal, characteristic, complemented
33	#72235865	C₂²	1	1	yes	characteristic, complemented
34	#72235054	C₂²	1	1	yes	characteristic, complemented
35	#72235054	C₂²	1	1	yes	characteristic, complemented
36	#72234825	C₂²	1	1	yes	characteristic, complemented
37	#72231822	C₂²	1	1	yes	characteristic, complemented
38	#72231822	C₂²	1	1	yes	characteristic, complemented
39	#72231810	C₂²	1	1	yes	characteristic, complemented
40	#72226038	C₂²	1	1	yes	characteristic
41	#72225994	C₂²	1	1	yes	characteristic, complemented
42	#72225873	C₂²	1	1	yes	characteristic, complemented
43	#72225733	C₂²	1	1	yes	characteristic
44	#72225733	C₂²	1	1	yes	characteristic
45	#72225658	C₂²	1	1	yes	characteristic
46	#72219034	C₂²	1	1	yes	characteristic, complemented
47	#72219022	C₂²	1	1	yes	characteristic, complemented
48	C₂×#131967	C₂²	1	1	yes	characteristic, complemented
49	C₂×#245225	C₂²	1	1	yes	characteristic
50	C₂×#245225	C₂²	1	1	yes	characteristic, complemented
51	C₂²×H(7,𝔽₂)	D₄	1	1	yes	characteristic, complemented
52	#8093110	D₄	1	1	yes	characteristic, complemented
53	#8093108	D₄	1	1	yes	characteristic, complemented
54	#7670333	D₄	1	1	yes	characteristic, complemented
55	#7670203	D₄	1	1	yes	characteristic, complemented
56	#7642273	C₂³	1	1	yes	characteristic
57	C₂×#35346	C₂³	1	1	yes	characteristic
58	C₂×#35346	C₂³	1	1	yes	characteristic
59	C₂×#35346	C₂³	1	1	yes	characteristic
60	C₂×#35346	C₂³	1	1	yes	characteristic
61	C₂×#34980	C₂³	1	1	yes	characteristic
62	C₂×#32898	D₄	1	2	yes	complemented
63	#6451874	D₄	1	2	yes	complemented
64	C₂×#20585	D₄	1	1	yes	characteristic, complemented
65	C₂×#20585	D₄	1	1	yes	characteristic, complemented
66	C₂×#20585	D₄	1	1	yes	characteristic, complemented
67	#612424	D₄	1	2	yes	complemented
68	#515285	C₂³	1	1	yes	characteristic
69	#449501	D₄	1	2	yes	complemented
70	#449475	D₄	1	2	yes	complemented
71	#417896	D₄	1	2	yes	complemented
72	#417824	D₄	1	2	yes	complemented
73	#392658	D₄	1	2	yes	complemented
74	#391709	D₄	1	2	yes	complemented
75	#382422	C₂³	1	1	yes	characteristic
76	#382422	C₂³	1	1	yes	characteristic
77	#382422	C₂³	1	1	yes	characteristic
78	#374004	D₄	1	2	yes	complemented
79	#359705	D₄	1	2	yes	complemented
80	#359428	D₄	1	2	yes	complemented
81	C₂×#10965	C₂³	1	1	yes	characteristic
82	C₂×#9495	D₄	1	2	yes	complemented
83	C₂×#9495	D₄	1	2	yes	complemented
84	C₂×#8547	C₂³	1	1	yes	characteristic
85	C₂×#8547	C₂³	1	1	yes	characteristic
86	C₂×#8541	C₂³	1	1	yes	characteristic
87	C₂×#8541	C₂³	1	1	yes	characteristic
88	#133278	D₄	1	4	yes	complemented
89	C₂³×H(5,𝔽₂)	C₂×D₄	1	1	yes	characteristic
90	C₂³×H(5,𝔽₂)	C₂×D₄	1	1	yes	characteristic
91	C₂³×H(5,𝔽₂)	C₂×D₄	1	1	yes	characteristic
92	C₂³×H(5,𝔽₂)	C₂×D₄	1	1	yes	characteristic, complemented
93	C₂⁵×D₄	C₂×D₄	1	1	yes	characteristic
94	C₂×#3490	C₂×D₄	1	1	yes	characteristic
95	C₂³×#120	C₂×D₄	1	2	yes	—
96	C₂³×#120	C₂⁴	1	1	yes	characteristic
97	C₂²×#518	C₂×D₄	1	1	yes	characteristic, complemented
98	C₂×#2849	C₂×D₄	1	1	yes	characteristic
99	C₂×#2849	C₂×D₄	1	1	yes	characteristic
100	C₂×#2846	C₂×D₄	1	1	yes	characteristic
101	C₂×#2846	C₂×D₄	1	1	yes	characteristic
102	#35390	C₂×D₄	1	1	yes	characteristic
103	#35390	C₂×D₄	1	1	yes	characteristic
104	#35367	C₂×D₄	1	1	yes	characteristic
105	#35367	C₂×D₄	1	1	yes	characteristic
106	#35346	C₂×D₄	1	2	yes	—
107	#35346	C₂×D₄	1	2	yes	—
108	#35346	C₂×D₄	1	1	yes	characteristic
109	#35346	C₂×D₄	1	1	yes	characteristic
110	#35346	C₂×D₄	1	1	yes	characteristic
111	#35346	C₂×D₄	1	1	yes	characteristic
112	#35346	C₂×D₄	1	2	yes	—
113	#35346	#41	1	2	yes	—
114	C₂×#2403	C₂×D₄	1	2	yes	—
115	C₂×#2403	C₂×D₄	1	2	yes	—
116	#34176	C₂×D₄	1	1	yes	characteristic
117	C₂×#2308	C₂×D₄	1	2	yes	—
118	C₂×#2308	C₂×D₄	1	1	yes	characteristic
119	C₂×#2308	C₂×D₄	1	1	yes	characteristic
120	C₂×#2308	C₂×D₄	1	1	yes	characteristic
121	C₂×#2308	C₂×D₄	1	1	yes	characteristic, complemented
122	C₂×#2308	C₂×D₄	1	1	yes	characteristic, complemented
123	C₂²×#379	C₂×D₄	1	1	yes	characteristic
124	C₂×#1819	C₂×D₄	1	2	yes	—
125	C₂×#1781	C₂×D₄	1	2	yes	complemented
126	C₂×#1781	C₂×D₄	1	2	yes	complemented
127	#14679	C₂×D₄	1	2	yes	—
128	#10965	C₂×D₄	1	2	yes	—
129	#10965	C₂×D₄	1	2	yes	—
130	#10965	C₂×D₄	1	2	yes	—
131	#10965	C₂×D₄	1	2	yes	—
132	#10965	C₂×D₄	1	2	yes	complemented
133	#8547	C₂×D₄	1	2	yes	complemented
134	C₂⁷	#120	1	1	yes	characteristic, complemented
135	C₂⁷	C₂²×D₄	1	1	yes	characteristic
136	C₂²×H(5,𝔽₂)	#120	1	1	yes	characteristic
137	C₂²×H(5,𝔽₂)	#120	1	1	yes	characteristic
138	C₂²×H(5,𝔽₂)	#120	1	2	yes	—
139	C₂²×H(5,𝔽₂)	#121	1	2	yes	—
140	C₂⁴×D₄	#120	1	2	yes	—
141	C₂⁴×D₄	#120	1	2	yes	—
142	C₂⁴×D₄	C₂²×D₄	1	1	yes	characteristic
143	C₂⁴×D₄	C₂²×D₄	1	1	yes	characteristic
144	C₂⁴×D₄	C₂²×D₄	1	1	yes	characteristic
145	C₂⁵×C₄	#120	1	1	yes	characteristic, complemented
146	C₂×#543	#120	1	1	yes	characteristic, complemented
147	C₂×#534	#120	1	1	yes	characteristic, complemented
148	C₂²×#120	C₂²×D₄	1	1	yes	characteristic
149	C₂²×#120	C₂²×D₄	1	1	yes	characteristic
150	C₂²×#120	C₂²×D₄	1	1	yes	characteristic
151	C₂²×#120	C₂²×D₄	1	1	yes	characteristic
152	C₂²×#120	C₂²×D₄	1	1	yes	characteristic
153	C₂²×#120	C₂×#41	1	1	yes	characteristic
154	C₂²×#120	H(5,𝔽₂)	1	1	yes	characteristic
155	C₂²×#120	H(5,𝔽₂)	1	1	yes	characteristic
156	C₂³×C₂²⋊C₄	#120	1	2	yes	—
157	C₂³×C₂²⋊C₄	C₂²×D₄	1	1	yes	characteristic
158	C₂³×C₂²⋊C₄	H(5,𝔽₂)	1	1	yes	characteristic
159	C₂³×C₂²⋊C₄	H(5,𝔽₂)	1	1	yes	characteristic
160	#2403	#120	1	2	yes	—
161	#2403	#120	1	2	yes	—
162	#2403	#121	1	2	yes	—
163	#2403	#121	1	2	yes	—
164	#2403	#121	1	2	yes	—
165	#2403	#121	1	2	yes	—
166	#2403	#127	1	2	yes	—
167	#2403	#127	1	2	yes	—
168	#2308	#120	1	2	yes	—
169	#2308	#120	1	1	yes	characteristic
170	#2308	#120	1	1	yes	characteristic
171	#2308	#121	1	2	yes	—
172	C₂×#379	#120	1	2	yes	—
173	C₂×#379	#120	1	1	yes	characteristic, complemented
174	C₂×#379	#120	1	1	yes	characteristic, complemented
175	C₂×#379	#120	1	4	yes	complemented
176	#2273	#120	1	1	yes	characteristic, complemented
177	#2273	#120	1	1	yes	characteristic, complemented
178	C₂×#342	#120	1	4	yes	complemented
179	C₂⁶	B(4,𝔽₂)	1	2	yes	—
180	C₂⁶	B(4,𝔽₂)	1	2	yes	—
181	C₂⁶	C₂×#120	1	1	yes	characteristic
182	C₂⁶	C₂×#120	1	1	yes	characteristic
183	C₂⁶	C₂×#120	1	1	yes	characteristic
184	C₂⁶	C₂×#120	1	1	yes	characteristic
185	C₂⁶	D₄²	1	1	yes	characteristic
186	C₂⁶	D₄²	1	1	yes	characteristic
187	C₂⁶	D₄²	1	1	yes	characteristic
188	C₂⁶	#546	1	1	yes	characteristic
189	C₂³×D₄	#392	1	2	yes	—
190	C₂³×D₄	#392	1	2	yes	—
191	C₂³×D₄	B(4,𝔽₂)	1	2	yes	—
192	C₂³×D₄	B(4,𝔽₂)	1	2	yes	—
193	C₂³×D₄	C₂×#120	1	1	yes	characteristic
194	C₂³×D₄	C₂×#120	1	1	yes	characteristic
195	C₂³×D₄	C₂×#120	1	1	yes	characteristic
196	C₂³×D₄	C₂×#121	1	1	yes	characteristic
197	C₂³×D₄	C₂×#121	1	1	yes	characteristic
198	C₂³×D₄	C₂×#121	1	1	yes	characteristic
199	C₂³×D₄	C₂×#127	1	1	yes	characteristic
200	C₂³×D₄	#534	1	1	yes	characteristic
201	C₂³×D₄	D₄²	1	1	yes	characteristic
202	C₂³×D₄	D₄²	1	1	yes	characteristic
203	C₂³×D₄	D₄²	1	1	yes	characteristic
204	C₂³×D₄	D₄²	1	1	yes	characteristic
205	C₂⁴×C₄	B(4,𝔽₂)	1	2	yes	—
206	C₂⁴×C₄	B(4,𝔽₂)	1	2	yes	—
207	C₂⁴×C₄	#534	1	1	yes	characteristic
208	C₂⁴×C₄	D₄²	1	1	yes	characteristic
209	C₂⁴×C₄	D₄²	1	1	yes	characteristic
210	C₂²×C₂²⋊C₄	C₂×#120	1	1	yes	characteristic
211	C₂²×C₂²⋊C₄	C₂×#120	1	1	yes	characteristic
212	C₂²×C₂²⋊C₄	#534	1	1	yes	characteristic
213	C₂²×C₂²⋊C₄	#534	1	1	yes	characteristic
214	C₂²×C₂²⋊C₄	#535	1	1	yes	characteristic
215	C₂²×C₂²⋊C₄	#535	1	1	yes	characteristic
216	C₂²×C₂²⋊C₄	D₄²	1	1	yes	characteristic
217	C₂²×C₂²⋊C₄	D₄²	1	1	yes	characteristic
218	C₂²×C₂²⋊C₄	D₄²	1	1	yes	characteristic
219	C₂²×C₂²⋊C₄	#546	1	1	yes	characteristic
220	C₂²×C₂²⋊C₄	#546	1	1	yes	characteristic
221	C₂²×C₂²⋊C₄	#546	1	1	yes	characteristic
222	#379	#392	1	2	yes	—
223	#379	#392	1	2	yes	—
224	C₂⁵	#2021	1	2	yes	—
225	C₂⁵	#2021	1	2	yes	—
226	C₂⁵	C₂×#392	1	1	yes	characteristic
227	C₂⁵	#2403	1	1	yes	characteristic
228	C₂⁵	#2403	1	1	yes	characteristic
229	C₂⁵	#2403	1	1	yes	characteristic
230	C₂⁵	#2408	1	1	yes	characteristic
231	C₂⁵	#2410	1	1	yes	characteristic
232	C₂⁵	#2410	1	1	yes	characteristic
233	C₂⁵	#2433	1	1	yes	characteristic
234	C₂⁵	#2846	1	1	yes	characteristic
235	C₂⁵	C₂×B(4,𝔽₂)	1	2	yes	—
236	C₂⁵	C₂×B(4,𝔽₂)	1	2	yes	—
237	C₂⁵	C₂×B(4,𝔽₂)	1	2	yes	—
238	C₂⁵	#3026	1	2	yes	—
239	C₂³×C₄	#2021	1	2	yes	—
240	C₂³×C₄	#2021	1	2	yes	—
241	C₂³×C₄	#2403	1	1	yes	characteristic
242	C₂³×C₄	#2408	1	1	yes	characteristic
243	C₂³×C₄	#2410	1	1	yes	characteristic
244	C₂³×C₄	#2410	1	1	yes	characteristic
245	C₂³×C₄	#2433	1	1	yes	characteristic
246	C₂³×C₄	#2613	1	1	yes	characteristic
247	C₂⁴	#14715	1	1	yes	characteristic
248	C₂⁴	C₂×#2021	1	2	yes	—
249	C₂⁴	#23895	1	1	yes	characteristic
250	C₂⁴	#23895	1	1	yes	characteristic
251	C₂⁴	#23900	1	1	yes	characteristic
252	C₂⁴	#23900	1	1	yes	characteristic
253	C₂⁴	#23900	1	2	yes	—
254	C₂⁴	#23900	1	1	yes	characteristic
255	C₂⁴	#23903	1	1	yes	characteristic
256	C₂⁴	#24404	1	2	yes	—
257	C₂⁴	#32898	1	2	yes	—
258	C₂³	#245225	1	1	yes	characteristic
259	C₂³	#245225	1	1	yes	characteristic
260	C₂³	#245246	1	1	yes	characteristic
261	C₂³	#245246	1	1	yes	characteristic
262	C₂³	#394058	1	1	yes	characteristic
263	C₂³	#394064	1	1	yes	characteristic
264	C₂³	#394066	1	1	yes	characteristic
265	C₂³	#511790	1	4	yes	—
266	C₂²	B(5,𝔽₂)	1	1	yes	characteristic
267	C₂²	#72218099	1	2	yes	—
268	C₂²	#72225609	1	1	yes	characteristic
269	C₂²	#72233445	1	1	yes	characteristic
270	C₂	#72255888	1	1	yes	characteristic
271	C₂	#72258791	1	1	yes	characteristic
272	C₂	#72265838	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²	#72225609
commutator, Frattini, cosocle	C₂³×#120	C₂⁴
LCS	G ▷ C₂³×#120 ▷ C₂⁴ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁵ ◁ C₂³×#120 ◁ G
LECS	G ▷ C₂³×#120 ▷ C₂⁴ ▷ C₂² ▷ C₁

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
1. #72266334	C₂	semidirect
2. #72266330	C₂	semidirect
3. #72266320	C₂	semidirect
4. #72266316	C₂	semidirect
5. #72266301	C₂	semidirect
6. #72266288	C₂²	semidirect
7. #72266245	C₂	semidirect
8. #72266233	C₂	semidirect
9. #72266215	C₂	semidirect
10. #72266201	C₂²	semidirect
11. #72266194	C₂²	semidirect
12. #72266177	C₂	semidirect
13. #72266142	C₂	semidirect
14. #72266064	C₂	semidirect
15. #72265775	C₂	semidirect
16. #72265766	C₂²	semidirect
17. #72259019	C₂²	semidirect
19. #72240423	C₂	semidirect
23. #72238594	C₂²	semidirect
25. #72237035	C₂²	semidirect
26. #72236921	C₂	semidirect
27. #72236901	C₂²	semidirect
28. #72236837	C₂²	semidirect
29. #72236816	C₂²	semidirect
31. #72236754	C₂²	semidirect
32. #72235872	C₂	semidirect
33. #72235865	C₂²	semidirect
34. #72235054	C₂²	semidirect
35. #72235054	C₂²	semidirect
36. #72234825	C₂²	semidirect
37. #72231822	C₂²	semidirect
38. #72231822	C₂²	semidirect
39. #72231810	C₂²	semidirect
41. #72225994	C₂²	semidirect
42. #72225873	C₂²	semidirect
46. #72219034	C₂²	semidirect
47. #72219022	C₂²	semidirect
48. C₂×#131967	C₂²	semidirect
50. C₂×#245225	C₂²	semidirect
51. C₂²×H(7,𝔽₂)	D₄	semidirect
52. #8093110	D₄	semidirect
53. #8093108	D₄	semidirect
54. #7670333	D₄	semidirect
55. #7670203	D₄	semidirect
62. C₂×#32898	D₄	semidirect
63. #6451874	D₄	semidirect
64. C₂×#20585	D₄	semidirect
65. C₂×#20585	D₄	semidirect
66. C₂×#20585	D₄	semidirect
67. #612424	D₄	semidirect
69. #449501	D₄	semidirect
70. #449475	D₄	semidirect
71. #417896	D₄	semidirect
72. #417824	D₄	semidirect
73. #392658	D₄	semidirect
74. #391709	D₄	semidirect
78. #374004	D₄	semidirect
79. #359705	D₄	semidirect
80. #359428	D₄	semidirect
82. C₂×#9495	D₄	semidirect
83. C₂×#9495	D₄	semidirect
88. #133278	D₄	semidirect
92. C₂³×H(5,𝔽₂)	C₂×D₄	semidirect
97. C₂²×#518	C₂×D₄	semidirect
121. C₂×#2308	C₂×D₄	semidirect
122. C₂×#2308	C₂×D₄	semidirect
125. C₂×#1781	C₂×D₄	semidirect
126. C₂×#1781	C₂×D₄	semidirect
132. #10965	C₂×D₄	semidirect
133. #8547	C₂×D₄	semidirect
134. C₂⁷	#120	semidirect
145. C₂⁵×C₄	#120	semidirect
146. C₂×#543	#120	semidirect
147. C₂×#534	#120	semidirect
173. C₂×#379	#120	semidirect
174. C₂×#379	#120	semidirect
175. C₂×#379	#120	semidirect
176. #2273	#120	semidirect
177. #2273	#120	semidirect
178. C₂×#342	#120	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
#72266334	2
#72266330	2
#72266320	2
#72266316	2
#72266301	2
#72266245	2
#72266233	2
#72266215	2
#72266177	2
#72266142	2
#72266064	2
#72265775	2
#72240423	2
#72236921	2
#72235872	2